Detector of dynamic gravitational force gradient fields

ABSTRACT

The invention is an instrument for detecting and measuring dynamic gravitational force gradient fields having a characteristic frequency and includes a body of elastic material having distributed mass including a dynamic mass quadrupole moment responsive to dynamic gravitational force gradients by the excitation of a mechanical vibrational resonant mode in the body of elastic material. The resonant mode has a vibration node in the body of elastic material whereat the body is supported. An electrical output is provided by means of at least one piezoelectric transducer coupled to the body which in conjunction with the aforementioned vibrational resonant mode forms an electromechanical circuit which is resonant at a selected frequency corresponding to the characteristic frequency of the dynamic gravitational force gradient.

United States Patent 1191 Weber et al. 1 Mar. 27,1973

[54] DETECTOR OF DYNAMIC 3,353,400 11/1967 Schafit ..73/24 G VITATI AL FR GRAD T RA ON 0 CE IEN OTHER PUBLICATIONS FIELDS An Article EntitledDetection and Generation of l [75] mentors x riz g z i gf i gf g.Gravitational Waves by J. Weber in Physical Review a vol. 117, No.1,.lan. 1, 1960 pp. 306-313.

[73] Assignee: Hughes Aircraft Company, Culver Primary Examiner-James J.Gill City, Calif. Attomey-James K. Haskell and Earnest F. Oberheim [22]Filed: Mar. 10, 1969 ABSTRACT [2]] Appl' 809464 The invention is aninstrument for detecting and mea- Related Application Data suringdynamic gravitational force gradient fields having a characteristicfrequency and includes a body of [63] Continuation of Ser. No. 524,312,Feb. 1, 1966,

abandoned, which is a continuation of Ser. No. 399,632, Sept. 28, 1964.

elastic material having distributed mass including a dynamic massquadrupole moment responsive to dynamic gravitational force gradients bythe excitation of a mechanical vibrational resonant mode in the body ofelastic material. The resonant mode has a vibration node in the body ofelastic material whereat the body is supported. An electrical output isprovided by means of at least one piezoelectric transducer coupled tothe body which in conjunction with the aforementioned vibrationalresonant mode forms an electromechanical circuit whichis resonant at aselected frequency corresponding to the characteristic frequency of thedynamic gravitational force gradient.

5 Claims, 11 Drawing Figures DETECTOR OF DYNAMIC GRAVITATIONAL FORCEGRADIENT FIELDS This application is a continuation of application Ser.No. 524,312 filed Feb. 1, 1966, now abandoned, which is a continuationof our copending application Ser. No. 399,632, DETECTOR OF DYNAMICGRAVITA- TlONAL FORCE GRADIENT FIELDS, filed Sept. 28,

This invention relates to gravitational field measurements andparticularly to an instrument for the detection and measurement ofdynamic gravitational force gradient fields.

Such dynamic gravitational force gradient fields can arise in many ways.One example is the time varying portion of the Newtonian gravitationalforce gradient field of an oscillating or rotating asymmetric mass.Another example is the gravitational radiation described by the Einsteintheory of gravity (General Theory of Relativity) that is emitted by anaccelerated mass quadrupole. Still example is the effective dynamicgravitational force gradient field that is created by the relativemotion of the detecting instrument through the static gravitationalforce gradient field of a mass.

The measurement of dynamic gravitational force gradients is ofimportance in technological and scientific areas. It is of greatimportance to technology to have an instrument that will measure theNewtonian gravitational force gradient fields existing around large orebodies and oil-containing formations while being operated on a movingplatform. It is also of great scientific importance to study thegravitational radiation emitted by astronomical sources such as rotatingbinary stars and exploding stars and galaxies to determine the innerstructure of these bodies, their dynamic behavior and the radiationgeneration mechanisms. It is of further scientific importance to studythe dynamic gravitational fields surrounding an oscillating or rotatingasymmetric mass to investigate the validity of Newton 's law of gravityin the high frequency region.

Prior to the devices described in the present invention, there did existdevices for the measurement of the anomalies created by geologicalformations. One instrument presently in use for measuring staticgravitational gradients is the Eotvos torsion balance which employs twoequal weights at different heights connected by a horizontal beam andsuspended by a torsion wire so that it is free to rotate in a horizontalplane about the'wire. The beam rotates only when a differentialhorizontal force acts on the weights, and this occurs when thegravitational field is distorted so that the horizontal component at oneend is different from that at the other. A number of measurements aretaken with the beam at different azimuthal orientations and the resultsare employed in equations which, when solved, provide a plurality ofquantities which define the gradient and curvature. The torsion balancehas only limited usefulness due to the length of time required to makemeasurements. This long measurement time is related to the inability toseparate the desired gravitational response due to the geologicalformation of interest from the noise sources arising from the operationof the balance and from the inherent noise of the instrument itself. Theuse of an instrument with dynamic response characteristics operated in amanner which creates a dynamic interaction between the instrument and agravitational field of the geologica] formations will create a dynamicinstrument response with frequency characteristics that allow thedesired signal to be separated from the noise by frequency filteringtechniques.

However, there did not exist in the prior art any adequate method formeasurement of the dynamic gravitational force gradient fields createdby sources of interest to the scientific community. There do existdevices for the measurement of the dynamic force fields created by theNewtonian gravitational attraction of rotating or oscillating asymmetricmasses. A survey of such work has recently published by J. C. Cook, OnMeasuring the Phase Velocity of an Oscillating Gravitational Field, J.Franklin Inst., 273, pp. 453- 471, (June 1962). Such devices usuallytake the form of an oscillatory pendulum (see J. C. Cook, FIG. 2). Thesependulum devices are force measuring devices and as such are not onlysensitive to the Newtonian gravitational force, but are also sensitiveto the inertial forces caused by rotations and vibrations. The field ofinterest in scientific work is the gradient of the dynamic gravitationalforce field. The gravitational radiation emitted by astronomical sourcesis of a force gradient or tensor type (rather than of a force or vectortype as is electromagnetic radiation) and therefore requires aninstrument that responds to dynamic gravitational force gradients. Thedynamic Newtonian gravitational fields surrounding an oscillatory orrotating asymmetric mass contain both force fields and force gradientfields, but the only part that can be unambiguously assigned togravitational effects is the dynamic force gradient, and therefore aninstrument that responds only to the force gradient and does not respondto the force itself is required in order to separate the desiredgravitational signal from the inertial noises.

Accordingly, it is an object of the present invention to provide aninstrument for the detection of geological formations from a movingplatform.

It is another object of this invention to provide an instrument for thedetection of dynamic gravitational force gradient fields.

Yet another object of the invention is to provide an instrument for thedetection of gravitational radiation.

Still another object of the present invention is to provide aninstrument for the measurement of the dynamic gravitational fieldssurrounding an oscillating or rotating asymmetric mass.

These and other objectives are achieved by a detector of dynamicgravitational force gradient fields according to the inventioncomprising a body of elastic material with distributed mass including amass quadrupole moment responsive to a dynamic gravitational forcegradient having a characteristic frequency. The body of elastic materialhas a mechanical vibrational resonant mode of relatively high Q and themass quadrupole response takes the form of relative accelerationsbetween portions of the distributed mass. Coupled to the body of elasticmaterial is an electrical output means which in conjugation with theaforementioned vibrational resonant mode includes a relatively high Qelectromechanical circuit which is resonant at a selected frequencycorresponding to the characteristic frequency of the dynamicgravitational force gradient. The electrical output means provides anoutput signal which is indicative of the magnitude and phase of thedynamic gravitational force gradient.

The invention and specific embodiments thereof will be describedhereinafter by way of example and with reference to the accompanyingdrawings, in which:

FIG. 1 is a schematic diagram of an embodiment of a dynamicgravitational force gradient field detector constructed according to theinvention;

FIG. 2 is a side view of one embodiment of a sensing element used in thedetector of FIG. 1;

FIG. 3 is a top and side view of a piezoelectric strain transducer whichwas used in the detector constructed as shown in FIG. 1;

FIG. 4 is a schematic diagram of an electrometer tube preamplifiercircuit shown in FIG. 1;

FIG. 5 is a side and end view of the sensing element of FIG. 2 supportedon a movable platform for convenient transportation in and out of avacuum chamber;

FIG. 6 illustrates schematically the interaction of a mass quadrupolewith gravitational radiation;

FIG. 7 is a schematic diagram of the coupling of two mass quadrupoles bydynamic gravitational fields; and

FIGS. 8, 9, 10 and 11 are perspective views of sensing elementsaccording to further embodiments of the invention.

The detection of dynamic gravitational force gradients fields isaccomplished in accordance with the present invention by following theconcept that any body of elastic material with distributed mass havingrelatively high Q mechanical vibrational resonant modes will react as asensing element to a dynamic gravitational force gradient field of thefrequency of the gravitational field is the same as the frequency of oneof the vibrational modes and the orientation of the vibrational mode andthe gravitational field is properly chosen.

The acoustic vibrations set up by the dynamic gravitational forcegradient field are no different than those set up by any otherdifferential force and the methods of studying the vibrations are wellknown in the field of acoustics.

To obtain the maximum coupling between the gravitational field and thevibrational mode, the system should have low internal losses (or highmechanical Q) and low external losses such as coupling of the materialto the surrounding air and supports.

Furthermore, any vibrating system alternates between storageof thevibrating energy in a kinetic form and a potentiaL form. The kineticenergy takes the form of motion, either longitudinal, transverse ortorsional. This motion occurs naturally at the free points in thevibrating system. The potential energy takes the form of compression,tension or shear of the material. These strains occur naturally at thenodes of the vibrating system. Since there is no motion of the vibratingsystem at a node, this is a preferred position to support the material.Also, it has been found that it is easier to detect the vibrations ofthe vibrating system by looking for the strains at the nodes rather thanthe more common and obvious method of looking for the motions at theantinodes. However, it may in certain instances be advantageous tocouple to the motions at the antinodes. This may be accomplished by theconventional use of capacitive, inductive and magnetostrictive devices.

To detect these strains, strain references (which do not detect motiondirectly) may be attached at the nodes where the strains are a maximum.Piezoelectric crystals have been found to be satisfactory straintransducers. By proper design and choice of crystal type, crystalorientation and electrical connections, the strain energy is convertedinto electrical energy by means of these transducers. These techniquesare well known in the field of acoustics and can be'found in referencesuch as Piezoelectric Crystals and Their Application to Ultrasonics byWarren P. Mason, D. Van Nostrand Co., Princeton, NJ. (1959).

The exemplary embodiment of a dynamic gravitational force gradient fielddetector shown in FIG. 1 illustrate the application of theabove-described concepts. An aluminum cylinder 11 is provided with aplurality of piezoelectric strain transducers 13 (see FIG. 2) whichdevelop alternating electrical signals corresponding to the strainspresent in the cylinder 11. The signals are coupled to a low noisepreamplifier 15 by means of leads 17 connected in parallel with asuperconducting coil 19 immersed in a liquid helium filled Dewar 21 areshunted by an air dielectric tuning capacitor 23 supported just outsidethe Dewar 21. The signal is further amplified by a narrow band high gainamplifier 25 coupled to the preamplifier 15 by cable 27. The amplifiedsignal is then provided from cable 29 to various conventional displaydevices such as the oscilloscope 31 for ac presentations and/or to a penand ink recorder 33 and data processing and storage equipment 35 througha detector-integrator 37 by means of cables 39, 41 and 43, respectively.

The aluminum cylinder 11, as shown in FIG. 2, represents a body ofelastic material with distributed mass including a mass quadrupole andis supported in the middle by a loop of wire 45. Because of its lowcost, ease of fabrication and high intrinsic Q, 6061 aluminum was usedas the cylinder 11 and operated in the first longitudinal vibrationalmode. The cylinder 11 was chosen to be 5 feet long and 2 feet indiameter with a first longitudinal mode frequency of 1657 cps and with ashallow groove (not shown) for the wire 45. In order to prevent thelowering of the interaction with the gravitational fields because ofcoupling of the cylinder 1 l to the air, the cylinder 1 l is placed in aconventional vacuum chamber which is not illustrated for the sake ofclarity. The chamber is maintained at a noncritical vacuum value of 500microns or better.

To detect the strains produced within the cylinder 1 1 by reaction witha dynamic gravitational force gradient field, the piezoelectric straintransducers 13 are attached to the cylinder 11 at the nodes where thestrains are a maximum. For the first longitudinal mode, this node occursat the middle of the cylinder 11'. The cylinder 11 is supported by theloop of wire 45 at this nodal line. Since there is a minimum amount ofmotion at the nodal line, there will be a minimum amount of interactionwith the supporting loop of wire 45 and this will minimize the nonlinearor dissipative interactions that lower the Q. The odd higher orderlongitudinal modes also have a node at the middle of the cylinder 11 sothat by changing the frequency of the external electronics to include ormatch the frequency of these odd higher harmonics the cylinder 11 can beused to detect dynamic gravitational force gradients at these otherfrequencies. The even order harmonics have an antinode at themiddleposition so therefore their measured Q is substantially less thanthe odd order harmonies. If it is desired to operate at an evenharmonic,

two or more supporting loops of wire would have to be used and placed atthe nodal points whose position would vary with the particular evenharmonic.

The transducers 13 are more clearly illustrated in FIG. 3. They arefabricated from X-cut quartz and are approximately one-fourth inch wide,2 inches long and 0.080 inches thick. The top and bottom portions arefirst plated with a very thin chromium base coat to wet the crystal andthen silver coatings 47 and 48, respectively, 0.0002 inches thick areused to act as terminals for the voltage developed by the quartz and asa means for connecting the leads 17 to the rest of the circuit. As shownin FIG. 3, the plating 48 on the bottom portion extends around endportion 49 of the transducer 13 to facilitate lead connection leaving agap portion 50 to prevent an electrical short circuit. The frequency ofthe voltage so developed by the transducer 13 will vary somewhat butwill be centered around the resonant frequency of the vibrating system(cylinder 11). In order that the electric output circuit (which includesamong others the transducer 13, the superconducting coil 19 and theinput impedance of the preamplifier l5) and the mechanical vibrationalresonant circuit of the cylinder 11 effectively combine to form anelectromechanical circuit with a single relatively high Q resonance, itis necessary to use a sufficiently large number of transducers toprovide a strong coupling. For the particular embodiment constructed andshown in FIGS. 1 and 2, 300 such transducers were used, although asomewhat lesser number may provide good results.

The piezoelectric transducers 13, because of their construction, willhave a capacitive reactance associated with them. It is necessary foroptimum signal tonoise ratio to tune out the reactance. This isaccomplished by shunting the parallel connected transducers 13 with aninductance. In this case, the inductance takes the form of the 1 Henrycoil 19. The value of inductance of the coil 19 is chosen so that theentire circuit, which includes the capacitance of the transducers 13,the stray capacitance and inductance of the interconnecting wires 17,the coil 19, the tuning capacitance 23 and the input impedance of thefollowing preamplifier stage 15, is resonant at the same frequency asthe vibrating system, i.e., 1,657 cps. This was easily accomplished bythe insertion of the small air dielectric capacitor 23 in parallel withthe coil 19 and tuned for a maximum output to the following stage whenthe cylinder 11 was mechanically excited by a very strong driving forceapplied to the supporting structure i.e., hitting the vacuum chamberwith a large hammer). The coil 19 consists of approximately 12,000 turnsof 3 mil double coated Niobiumwire wound on a hollow quartz coil from(not shown) which is 3 inches long and 2 inches in diameter. In order toreduce unwanted Johnson noise to be discussed later), the coil 23 ismade superconducting by its immersion in the liquid helium bathcontained in the 25 liter Dewar 21.

To prevent heavy loading of the tuned circuit, the preamplifier shouldhave a very high input impedance of the order of 100 megohms and more.Also, the preamplifier 15 should have a low noise figure to prevent themasking of theweak signal. The combination of a high input impedance anda low noise figure with a very high input impedance is obtainable, forexample, by the use of an electrometer tube circuit such as the oneshown in FIG. 4. Here, the leads 17 couple the input signal developed bythe transducers 13 to the grid electrode 51 of electrometer tube 53.Proper filament voltage and operating bias are provided by a droppingresistor 55 connected between a 42 volt source (not shown) of B+ voltageand filament terminal 57, and a parallel resistor 59-capacitor 61combination connected between the other filament terminal 63 and acommon ground return 65. A filament bypass capacitor 66 is connectedbetween the filament terminals 57 and 63. The plate load resistors 67connects B+ to an anode terminal 69 of the tube 53. The input signal isamplified by the amplifier 15 by a factor of approximately 2, theamplified output signal of which is coupled to the narrow band amplifier25' by capacitor 73 and the cable 29. The value for the variouscomponents of the preamplifier 15 is given in the following table:

53 Type 5886 electrometer tube 55 3,750 ohm rwatt carbon resistor 59 300ohm zwatt carbon resistor 61 20 p.f50 volt capacitor 66 50 pf 10 voltcapacitor 67 150,000 ohm 1 watt carbon resistor 73 0.1 pfcapacitor- Thepreamplifier 15 may be immersed in the liquid helium bath in the Dewar21 to help reduce noise even more. The electrometer tube circuit hasbeen operated at liquid helium temperatures (4.2K) with verysatisfactory results.

The signal from the preamplifier 15 is amplified by the high gain(approximately 10,000), turned amplifier 25 which has a center frequencyof 1,657 cps. Am,- plifiers capable of this type operation are wellknown in the art use tuned RC feedback networks at each amplifyingstage. to narrow the bandpass to reduce the effects of wide band noise.Also, twin-tee networks are used to reject 60 cps hum. A Strandberg typeamplifier may be used. Such an amplifier is described in a paperentitled Recording Magnetic-Resonance Spectrometer, by M .W.P.Strandberg, 'et (11., in the Review of Scientific Instruments, 27,August 1956, p. 604, FIG.' 7, with the RC constants in the feedbacknetworks chosen to resonate at 1,657 cps.

The signal developed by the transducers 13 will not be very large inmagnitude and it is thus advisable to consider the electrical andmechanical sources of noise that could possibly mask the desired signal.

Noise can originate in the vibrating systemin two ways. By 1) thermalfluctuations and by (2) mechanical couplings. The vibrating system isdesigned so that the mechanical coupling of the important vibrationalmode to the other mechanical parts of the vibrating system is eliminatedor minimized. Thisis a matter of good design that will depend upon theparticular vibrating system. The techniques of isolating degrees offreedom of mechanical systems are well known, the basic principle beingthat all necessary support forces should be applied at nodal points andthat the materials used be strong enough so that the strains remainlinear (Hookes Law remains valid). If nonlinear strains are allowed todevelop, then undesired coupling into the vibrational mode becomes moreprobable.

To eliminate the coupling of the external acoustic noise sources to thevibrating system through the supports, it is sufficient to design thesupports and the electrical leads so that they act as an acoustic filterto prevent acoustic vibrations from reaching the vibrating system. Thesetechniques are well known in acoustic engineering. For connections undertension (the leads 17, for example), the necessary impedance mismatchesare created by adding masses such as lead blocks (not shown) at properpoints. The areas with increased mass will have a different acousticimpedance than the neighboring areas and the acoustic vibrations will bereflected back away from the vibrating system.

For connections under compression, the necessary impedance mismatchesare created by constructing the connections of alternating layers ofmaterials with different acoustic impedances such as metal and rubber.For example, FIG. illustrates an exemplary manner of supporting thecylinder 11 on a movable vehicle 81. Here, /4 inch rubber pads 83separate blocks of steel 85 of various dimensions to support a crossbeam 87 to which wire 45 is attached. The acoustic vibrations will bereflected at the interfaces of the two differing materials and will bekept away from the vibrating system. The design of the filter isstrongly dependent upon the primary function of the connector and thefrequency of the vibrating system. But, these are common problems inacoustic engineering and the solution is straight-forward once the formof the device is decided upon.

Noise can also originate in the electronic circuitry. The common sourcesare nonlinear effects in the materials used in the circuitry and theJohnson noise given off by an resistance in the circuitry. To minimizethe effect of this noise, the coil 19 is preferably in a superconductingstate. This not only eliminates a major source of noise but alsoincreases the Q of the circuit. The coil 19 may also be encased in alead can (not shown) which is in a superconducting state andelectrically connected to a common ground side 88 (FIG. 1) of the coil19 to form an electromagnetic shield. This will even further reduce thenoise.

Still another source of noise is the preamplifier stage due to theresistive component of the input impedance which creates Johnson noise,and to the random emission of electrons in the tube which causes shotnoise. To decrease the noise due to the input resistance, an amplifierhaving an input resistance which is larger than the parallel resistanceof the electronic circuitry may be used. To minimize the shot noise, itmay be desirable to use an amplifier having a good noise figure. In thedetector constructed according to the invention, it was found that asimple electrometer tube circuit such as shown in FIG. 4 was quitesatisfactory for this service.

In order to understand the method of operation of the devices of thepresent invention and to demonstrate that the interactions observed aredue to coupling of the devices to dynamic gravitational fields ratherthan some other interaction, a brief outline of the theory of dynamicgravitational, interactions is here presented.

A dynamic gravitational field is defined as the time varying componentof the gravitational interaction between two structures which are inrelative motion. This is usually understood to mean that one of thebodies is undergoing oscillatory or translational motion in inertialspace and therefore its gravitational field varies with time. This timevarying gravitational field will then exert time varying forces on adetecting device.

There is also another possible method for a dynamic gravitationalinteraction in which the source body is stationary with respect toinertial space, and its gravitational field does not vary with time, butonly with position. If the detecting device is moving with respect toinertial space, then the spatially varying gravitational field of thesource is transformed, in the detecting bodys frame of reference, into atime varying gravitational field.

Mechanically, the two are nearly equivalent although the second type ofdynamic gravitational interaction is usually more practical.

It shall be assumed that all gravitational effects are correctlydescribed by Einsteins theory of gravity (General Theory of Relativity).(See for example A. Einstein, The Meaning of Relativity, 5th Edition,Princeton University Press, Princeton, NJ. (1955); C. Moller, The Theoryof Relativity, Oxford University Press, London (1957); or J. Weber,General Relativity and Gravitational Waves, Interscience Publications,Inc., N.Y. (1961).) It shall also be assumed that the cosmologicalconstant, sometimes included in the theory, is too small to be ofinterest in experimental work so that the field equations of generalrelativity will be assumed to have the form:

jigg /gm 8"1rG [C iozB (1) where c= 3.00 X 10 m/sec is the speed oflight,

G 6.67 X 10- m lkg see is the Newtonian constant of gravity Tu is thestress-energy-momentum tensor gaB is the metric tensor describing theproperties of gravitation and space which is defined by the square ofthe interval ds along the space-time world line s -ds =g ;-1 dx dx 2With the signature chosen so that the flat space metric has the form:

R is the curvature scalar obtained from the contraction of the Riccitensor.

and R y is the Ricci tensor obtained from the contraction of the Riemanntensor.

017,1! all/Y 5B a7 57 11/3 and the Christoffel symbols are defined interms of the metric tensor the Ricci tensor (5) and the curvature scalar(4) are sums of the products of the metric tensor and the Riemanntensor, this means that the left hand sides of the field equations (1)comprise a very complicated, nonlinear, second order partialdifferential prescription for the calculation of the components of themetric tensor, given the distribution and behavior of matter and energyin the form of the stress-energy-momentum tensor on the right handsides.

The usual process of calculating the dynamical behavior of a systemunder the influence of gravitational and other forces is quitecomplicated. First, all the mass and energy in both the system beinginvestigated and in the sources of the dynamic fields must bedetermined. Then, using these in a prescribed manner, the 10 componentsof the stress-energy-momentum tensor are calculated. Next, using thestressenergy-momentum tensor as the source term in the field equations(1), these 10 nonlinear differential equations for the 10 components ofthe metric tensor are solved. Then the metric tensor is used in thegeneralized equations of motion to determine how the system behaves.

For experimental purposes, it is not necessary to use the full fieldequations. The gravitational forces available are nearly always weakenough so that the nonlinear terms in the field equations arenegligible. Often the velocities involved are small enough so that evenspecial relativistic effects can be ignored. Thus, it is only necessaryto carry out the calculations using an appropriate approximation to thefull field equations.

To obtain a simplified form of the Einstein field equations that issuitable for experimental work, the weak field approximation will beused (see Weber, p. 87ff). This approximation uses the assumption thatthe gravitational potential energy in the gravitational fields involvedin an experiment is small compared to the kinetic energy and the restenergy of the masses and nongravitational fields used in the experiment.This assumption is satisfied to a very high degree of approxi mation byall conceivable experimental situations.

If the gravitational fields are weak, then the metric tensor can beapproximated by gas e 018+ has (9) where 6113 is the flat space metricgiven previously and the h 043 are the perturbations of the metricintroduced by the masses generating the gravitational fields. If thetensor gravitational potential is now defined as a certain combinationof the perturbations afl on the flat spacemetric tensor (fi B= oB /Z Qh(l0) and the necessary substitutions are made, we find that thenonlinear Einstein field equations become linear Poisson equations,orthe weak field equations [:1 dg q g where I] is the dAlembertianoperator.

In the field equations, the stress-energy-momentum tensor is the meansof coupling to the gravitational field. The various components of thestress-energy-momentum tensor are related to the density and momentum ofthe mass-energy involved in the device of the invention.

The stress-energy-momentum tensor is known for a number of differentsources.

The stressenergy-momentum tensor of physical matter is Ta =p.c UaUB+Safl (17) where p. is the density of the matter. S dB is the elasticstress tensor of the material and U... is the four velocity defined by UE (l/ )(dx/dr) dt/d-r(l.(V/C)) (18) where v is the physical velocity.Thus, mechanical stresses can couple to gravitational fields throughtheir contribution to the stress-energy-momentum tensor. If the energyand momentum in the stress or pressure fields can be neglected incomparison to the rest mass energy and momentum of the masses involved,then the stress-energy-momentum tensor has the simplified form T ,u.c U"U 20 so that oscillating masses can couple to dynamic gravitationalfields through their contribution to the stress-energy-momentum tensor.Because gravitational experiments nearly always involve the use ofphysical masses with their large amounts of concentrated energy density,it is this final form of the stress-energy-momenturn tensor that isusually used for calculations. However, under certain conditions andespecially for high frequency operation, the mechanical stresscontributions become as important as the mass motion contribution.

The simplest approximation to the weak field equations assumes that theonly sources of gravitational effects are physical masses and that themasses involved not only have weak gravitational fields, but they alsohave low rotational rates or velocities compared with the speed oflight. In this approximation, the only component of thestress-energy-momentum tensor (20) that is not negligible is 21) Sincethe velocities are assumed to be low, the time derivatives of thepotential are smaller than the spatial gradients of the potential so theweak field equations (1 1) reduce to The equation for the time-likecomponent of the tensor gravitational potential (22) is a threedimensional Poisson equation which has the solution This component ofthe tensor gravitational potential is easily seen to be directly relatedto the scalar potential so that, as expected, the Einstein gravitationalfield equations reduce to the Newtonian gravitational field equation inthe lowest approximation.

Normally, the interaction of the Newtonian gravitational field with adetecting mass is considered as a purely static one, but if the positionof the source mass (or the detecting mass) changes, then thegravitational field will vary with time and the interaction becomes adynamic one. I

Besides the dynamic Newtonian interaction,there is also another dynamicinteraction governed by the Einstein law of gravity. This isgravitational radiation. The behavior of such radiation in anyphysically realizable experiment is governed by the weak fieldapproximation to the field equations, for as it stands, the weak fieldapproximation is a wave equation for the tensor gravitational potential(b g. The velocity of propagation is the same as the velocity of light.

The solution of the weak'field wave equations with a nonvanishing sourceterm is well known as:

where r is the field point, r is the source point and R Ir r' '18 thedistance from the source pomt to the field point.

The straightforward method of finding the solutions to these equationsis to calculate the kinetic and stress energy in the source and usethese directly. However, because of the laws of conservation of energyand momentum the various components of the stress-energymomentum tensorare not independent and it is possible to convert the integrals over thestresses into integrals over the more easily identified motional energyof the sources. When we do this, we see that the spatial components ofthe tensor gravitational potential depend upon the second timederivatives of the mass quadrupole moment of the source This equationshows that the lowest mode of gravitational radiation possible isquadrupole radiation. Thus, in general, any accelerated (i.e., rotatingor vibrating) mass quadrupole will emit gravitation radiation.

The fact that gravitational radiation is quadrupole can also beunderstood in terms of the law of conservation of momentum. In anysystem of particles, the momentum of these particles must be conserved.

mg m 30. .=0 27 But the gravitational radiation that is possible fromthese masses must come from the acceleration of the masses and if theequation for the conservation of momentum is differentiated, then m +m:t =O 28 so that the gravitational radiation from each part of thesource is cancelled (to first order) by the gravitational radiation fromsome other part of the source. Thus, there is no dipole gravitationalradiation, only quadrupole or higher multipole radiation.

The simplest quadrupole mass source for the calculation of gravitationalradiation energy emission is two equal masses rotating about theircenter of mass.

An example of such a rotating source would be a binary star system.Because the binary system is held together by gravitational forcesrather than mechanical forces, it can emit copious amounts ofgravitational radiation and this radiation can be detected by thedevices of the invention.

' Gravitational radiation can be thought of either as a propagatinggravitational field or the propagation of the curvature of space-time.This radiation, .be it space curvature or gravitational field, willexert forces on objects with mass. Since gravitational radiation and alldynamic gravitational interactions are of quadrupole nature because ofthe conservation of momentum, it is necessary to use at least a massquadrupole to interact with the radiation in order to detect itsexistence.

A mass quadrupole, by its very nature, involves a length. It is notdefined at a point but exists over a region about some point inspace-time. Since the masses 1 or mass density making up the quadrupolemust be at different points in space-time, they each follow their ownseparate equation of motion along their own world line. Then, if thereare any gradients in the gravitational field or space curvature due togravitational radiation, the paths of the two parts of the massquadrupole will differ slightly, indicating the presence of theradiation.

Conceivably, the two particles necessary for the mass quadrupole couldbe in free fall (connected only by their gravitational attraction), thenthe passage of gravitational radiation would cause relative motionbetween the two particles. But then there are difficulties as to whetherthe particles would be able to extract energy from the radiation orwhether'they would just return to rest after the passage of theradiation.

If, however, the two parts of the mass quadrupole are coupled with anenergy converting mechanism that transforms the stress energy introducedby the gravitational radiation into some other form of energy such asacoustic vibrations or thermal energy, then the energy, once convertedby these irreversible processes, cannot be completely reconverted againinto gravitational energy. Thus, the radiation can be detected byextracting some of the energy out of the wave using a mass quadrupoleand an energy conversion mechanism.

There still might be some doubt as to whether the stresses due to thegravitational radiation are real and can exert strains in a materialbody. For example, the

special relativistic contraction due to high relative motion is not aphysical effect that can be sensed by the rapidly moving object, and itmight be argued that because of the principle of equivalence betweengravitational fields and coordinate systems a similar effect wouldhappen with gravitational radiation. However, the principle ofequivalence is only valid at a point, and a mass quadrupole does notoperate over a point, so that although the acceleration of the center ofmass of the mass quadrupole cannot be observed locally, the gradient ofthe acceleration can be observed by the relative acceleration of the twomasses of the mass quadrupole. The reality of the tides is an excellentexample: they are purely gravitational in nature, but the coordinatesystem that nature chooses to use for the motion of the earth has onlyfound a way to remove the center of mass forces and has not found a wayto compensate for the dynamic gradient forces; they are real and energycan and is being extracted from them.

If a mass quadrupole is used for the detection of gravitationalradiation, then there is present two particles, each with its ownequation of motion, and coupled together by their mutualnongravitational forces. The behavior of such a system is described bythe equation of differential motion (see Weber, p. 124ff) D n" D F5 8 vs aU'y B d 33 1 D5 U n Du) mc w r mc where U is the four velocity, F isthe nongravitational forces coupling the two parts of the massquadrupole and D/Ds is the covariant derivative with respect to the times, n is the spatial displacement of the mass points, f is the forcedifference due to the spatial gradient of the force DF /Dw operatingacross the differential distance dw, and Ri is the Riemann curvaturetensor.

The use of this equation may be examined for the very simple massquadrupole detector consisting of two masses, each of mass m and aspring (see FIG. 6). The two world lines s of interest are those throughthe centers of two identical masses 101 and the distance n between thetwo world lines consists of the initial length r of a spring 102 whichdoes not vary with time and a small time varying extension y Thenongravitational forces connecting the two masses 101 consists of arestoring spring force kff plus dissipation d; due to the motion of thespring 102.

In the limit of small, nonrelativistic vibrations of a freely fallingdetector, the equation of differential motionbecomes:

dZga

The previous description of the operation of the devices of the presentinvention in terms of the Einstein theory of gravity is informative andimportant in terms of the utilization of the devices in communication bygravitational radiation since only the Einstein theory can adequatelydiscuss the radiation aspects of gravitation. However, for a detailedquantitative discussion of the operation of the devices of theinvention, it is only necessary to assume a very simplified model thatis completely describable by the Newton law of gravity.

The dynamic gravitational interaction of two oscillating massquadrupoles may be investigated and are shown in FIG. 7. Each massquadrupole 151, 153 consists of two masses 155 and 157 having a mass ofm connected by spring 159 and 161, respectively, of nominal length 2!,spring constant k, and damping constant D constrained to move only alongthe axis 162 and 163 through the two mass centers 165, 167. If it isassumed that the two mass quadrupoles 151, 153 are lying parallel toeach other, with separation d and one (the generator) is being driven byan energy source (not shown) so that the masses 155, for example, areundergoing a periodic displacement a cos mt, then the gravitationalfield of the generator will contain periodic variations. These periodicvariations of the gravitational field will cause periodic forces F andF, to be exerted on the two masses 157 of the second mass quadrupole(the detector) and will cause it to respond with a periodic motion withrelative amplitude 5.

If Newtons law of reaction (F mA) is used for the masses, then thedifferential acceleration of the two masses 157 of the detector causedby the differential forces of the masses 155 is given by Gmma cos wtGmm(2l a cos wt) (d a cos wt) [d (21+ a cos wz) where the first term isdue to the differential forces caused by the coupling F of the detectormasses 157 to the nearest generator masses 155 and the second term isdue to the differential forces caused by the coupling F between thedetector masses 157 and the furthest generator masses 155 (neglectingself interactions).

If I d .5 is assumed, then after simplification there is obtained anequation k D 26ml g m m (d +4-l C0swt+. (38) 1f the mass m and springconstant k are chosen so that the detector mass quadrupole 153 isresonant at the frequency w of the generator 151, and if the dampingconstant is expressed in terms of the frequency w and energy storagefactor Q, then the equation for the detector becomes 5+ (Mam =(Gma/d cos(of (39) To obtain a quantitative estimate of the coupling to beexpected from a practical physical embodiment of this simplified model,the following assumptions are made.

m 10 kg Q 10 (easily obtainable in mechanical systems) 11/! 10' (easilyobtainable for transverse vibrations) m=21rX16Ocps=10 d 1 foot.

With these experimental parameters, the strain predicted by theNewtonian laws of force and gravitation is e 2 X 10' There existcommercially, barium titanate dynamic strain transducers which have avoltage-strain characteristic of 1.6 X 10 volts per unit strain. Thevoltage output of such a transducer coupled to a strain of 2 X in themass quadrupole detector is 3.2 microvolts. This is a voltage that iseasily measured by standard voltage measurement devices.

The above discussion shows that signals can be transmitted between twomass quadrupoles by dynamic gravitational interactions and demonstratesthat the devices of the invention couple to the dynamic gravitationalfields in a practical and usable manner, which is completely describablein terms of the well-verified Newtonian laws of gravity as well as theEinstein theory of gravity, and shows that the observed coupling is dueto dynamic gravitational interactions and not due to acoustic couplingor other extraneous factors.

With reference to FIG. 8, there is shown a further embodiment of a bodyof elastic material with distributed mass including a mass quadrupoletaking the form of a relatively long thin aluminum bar 201 having anodal line 203 located at the center of the bar 201. This embodimentdeals with transverse vibrations of the bar 201. The bar 201 may besupported at the nodal line 203 by a wire (not shown). Straintransducers, such as piezoelectric transducers 205, may be attached atthe nodal line 203 or at convenient points adjacent to this line such asshown in FIG. 8. The transducers 205 are connected to the remainder ofthe circuit by wires 17 in the same manner as shown in FIG. 1.

In contrast to the embodiment of the invention as shown in FIG. 1 and 2where tensions induced by dynamic gravitational force gradients aresensed, this embodiment is sensitive to torques induced by such forcegradient fields. It should be noted that quartz may be substituted forthe aluminum making up the bar 201 (or cylinder 11) which will providean even higher-Q.

Inorder to reduce the physical dimensions of the sensing element such asthe bar 201 and still obtain a vibrational mode having a relatively longvibrational period to facilitate the measurement of low frequencygravitational force gradients, the mass of sensing element may beconcentrated, at points closer to each other than would be possible withan elongated bar of constant cross section with the same total mass.This embodiment is shown in FIG. 9 where a sensing element 251 comprisesa relatively short bar portion 253 and concentrated mass portions 255 atthe ends of the bar portion 253. The element 251 has a nodal line 257 atthe center of the bar portion 253 where the element 251 may be supportedin the same manner as shown in FIGS. 1, 2 and 5. This embodiment, likethe one shown in FIG. 8, is sensitive to torques induced by dynamicgravitational force gradient fields.

With reference to FIG. 10, there is shown an X configuration sensingelement 301 that provides an embodiment which has better dynamic balancecharacteristics since it interacts less with the supporting structure.Here, the sensing element 301 comprises, in effect, two orthogonallydisposed bar sections 303 and 305, respectively, joined at their centers307 where the line of intersection is along the width of the bars 303and 305. However, in order to assure cooperative interaction between thebars 303 and 305, it is advisable to provide fillet sections 311 wherethe bars 303 and 305 are joined. To sense torques in the plane of thesensing element 301 produced by dynamic gravitational force gradientfields, electromechanical strain transducers 313 are attached byconventional adhesive techniques to each side of the bars 303 and 305adjacent the fillet sections 311. The nodal lines for the two orthogonalbar sections coincide at their centers 307, where ideally, thetransducers 313 should be placed. But, in the practical embodiment,these transducers may be placed as close to the nodal lines as ispossible.

FIG. 11 illustrates an embodiment which is sensitive to torques normalto the plane of the sensing element. From this figure, it can be seenthat the sensing element 401 is similar in many respects to the element301 in FIG. 10 except that bar sections 403 and 405 are joinedorthogonally at their centers 407 where the line of intersection isalong the thickness .of the bars 403 and 405. Here, transducers 413 areattached to each bar on each side of the center line 407 and areconnected to the remainder of the detector circuitry as shown in FIG. 1.

From the foregoing, it will be seen that there is described a dynamicgravitational force gradient field detector which may be advantageouslyused, for example, as an instrument for the detection of geologicalformations, and when using certain described sensing elements, may besmall enough to be transported in a moving platform such as an airplanefor large ore geological surveying.

Although several specific embodiments have been herein illustrated, itwill be appreciated that other organizations of the specificarrangements shown may be made within the spirit and scope of theinvention. Additionally, other components or elements may be substitutedfor those which have been particularly named. For example, the sensingelement for the detector can have a torsional vibrationalresonant modeand/or a shear vibrational resonant mode which would be excited by thetorques due to the dynamic gravitational force gradients. Furthermore,since every elastic body has a multitude of mechanical vibrationalresonant modes, each having a characteristic frequency, then each modecan be coupled into a separate electrical circuit having a correspondingresonant frequency to provide more information on the frequency spectrumof the dynamic gravitational force gradient field exciting the sensingelement. Yet another possibility is to place a number n of the describeddetectors in a region whose linear dimensions are less than half agravitational wavelength. The electrical outputs of the 1: number ofdetectors may then be combined together in a coherent manner to providea capture cross section which is proportional to the it rather than 11.Finally, it should be clearly understood that the embodiments of theinvention described in the specification will function advantageouslywithout the elaborate means described herein to reduce the noise factorbut, of course, with reduced sensitivity.

Accordingly, it is intended that the foregoing disclosure and thedrawings shall be considered only as illustrations of the principles ofthis invention and are not be construed in a limiting sense.

What is claimed is:

l. A detector of dynamic gravitational force gradient fields,comprising: a body of elastic material with distributed mass including adynamic mass quadrupole moment responsive to a dynamic gravitationalforce gradient having a characteristic frequency, said response takingthe form of relative acceleration between portions of said distributedmass, said body of elastic material having a relatively high Qmechanical vibrational resonant mode excited by said relativeacceleration, said mode having a vibration node in said body; meanscoupled to and supporting said body at substantially said vibrationalnode; isolation means coupled to said body of elastic material forfiltering out all excitations except said gravitational fields; andelectrical output means including at least one piezoelectric transducercoupled to said body of elastic material adjacent said vibrational nodeto form a relatively high Q electromechanical circuit which is resonantat said characteristic frequency of said dynamic gravitational forcegradient for providing an output signal.

2. A detector of dynamic gravitational force gradient fields,comprising: a body of elastic material with uniformly distributed massincluding a dynamic mass quadrupole moment responsive to a dynamicgravitational force gradient having a characteristic frequency, saidresponse taking the form of relative acceleration between portions ofsaid distributed mass, said body of elastic material having a relativelyhigh Q longitudinal mechanical vibrational resonant mode excited by saidrelative acceleration, said mode having a nodal line around the centerof said body; isolation means coupled to said body of elastic materialfor filtering out all excitations except said gravitational fields; andelectrical output means including at least one piezoelectric transducerattached to said body adjacent said nodal line to form a relatively highQ electromechanical circuit which is resonant at said characteristicfrequency for providing an output signal.

3. A detector of dynamic gravitational force gradient fields,comprising: a body of elastic material of uniformly distributed massincluding a dynamic mass quadrupole moment responsive to a dynamicgravitational force gradient having a characteristic frequency, saidresponse taking the form of relative acceleration between portions ofsaid distributed mass, said body of elastic material having a relativelyhigh Q longitudinal mechanical vibrational resonant mode excited by saidrelative acceleration, said mode having a nodal line around the centerof said body; isolation means coupled to said body of elastic materialfor filtering out all excitations except said gravitational fields; andelectrical output means including at least one piezoelectric transducerattached to said body adjacent said nodal line to form a relatively highQ electromechanical circuit including a superconducting coil and atuning capacitor which circuit is resonant at said characteristicfrequency for providing an output signal.

4. A detector of dynamic gravitational force gradient fields,comprising: a body of elastic material supported by a steel wire, saidwire in turn being supported by an acoustically filtered supportstructure, said body comprising a distributed mass including a dynamicmass quadrupole moment responsive to a dynamic gravitational forcegradient having a characteristic frequency, said response taking theform of relative acceleration between portions of said distributed mass,said body having a relatively high Q longitudinal mechanical vibrationalresonant mode excited by said relative acceleration, said mode having anodal line at the position of said wire; and electrical output meansincluding a plurality of parallel connected piezoelectric transducersattached to said body adjacent said nodal line to form a relatively highQ electromechanical circuit that is resonant at said characteristicfrequency for providing an output signal.

5. A detector of dynamic gravitational force gradient fields,comprising: a body of elastic material supported by a steel wire, saidwire in turn being supported by an acoustically filtered supportstructure, said body comprising a distributed mass including a dynamicmass quadrupole moment responsive to a dynamic gravitational forcegradient having a characteristic frequency, said response taking theform of relative acceleration between portions of said distributed mass,said body having a relatively high Q mechanical vibrational resonantmode excited by said relative acceleration, said mode having a nodalline at the position of said said wire; and electrical output meansincluding a plurality of parallel connected piezoelectric transducersattached to said body adjacent said nodal line to form a relatively highQ electromechanical circuit which circuit includes a superconductingcoil encased in a superconducting shield disposed in a liquid heliumbath and a tuning capacitor tuned so that said circuit isresonant atsaidcharacteristic frequency for providing an output signal.

1. A detector of dynamic gravitational force gradient fields,comprising: a body of elastic material with distributed mass including adynamic mass quadrupole moment responsive to a dynamic gravitationalforce gradient having a characteristic frequency, said response takingthe form of relative acceleration between portions of said distributedmass, said body of elastic material having a relatively high Qmechanical vibrational resonant mode excited by said relativeacceleration, said mode having a vibration node in said body; meanscoupled to and supporting said body at substantially said vibrationalnode; isolation means coupled to said body of elastic material forfiltering out all excitations except said gravitational fields; andelectrical output means including at least one piezoelectric transducercoupled to said body of elastic material adjacent said vibrational nodeto form a relatively high Q electromechanical circuit which is resonantat said characteristic frequency of said dynamic gravitational forcegradient for providing an output signal.
 2. A detector of dynamicgravitational force gradient fields, comprising: a body of elasticmAterial with uniformly distributed mass including a dynamic massquadrupole moment responsive to a dynamic gravitational force gradienthaving a characteristic frequency, said response taking the form ofrelative acceleration between portions of said distributed mass, saidbody of elastic material having a relatively high Q longitudinalmechanical vibrational resonant mode excited by said relativeacceleration, said mode having a nodal line around the center of saidbody; isolation means coupled to said body of elastic material forfiltering out all excitations except said gravitational fields; andelectrical output means including at least one piezoelectric transducerattached to said body adjacent said nodal line to form a relatively highQ electromechanical circuit which is resonant at said characteristicfrequency for providing an output signal.
 3. A detector of dynamicgravitational force gradient fields, comprising: a body of elasticmaterial of uniformly distributed mass including a dynamic massquadrupole moment responsive to a dynamic gravitational force gradienthaving a characteristic frequency, said response taking the form ofrelative acceleration between portions of said distributed mass, saidbody of elastic material having a relatively high Q longitudinalmechanical vibrational resonant mode excited by said relativeacceleration, said mode having a nodal line around the center of saidbody; isolation means coupled to said body of elastic material forfiltering out all excitations except said gravitational fields; andelectrical output means including at least one piezoelectric transducerattached to said body adjacent said nodal line to form a relatively highQ electromechanical circuit including a superconducting coil and atuning capacitor which circuit is resonant at said characteristicfrequency for providing an output signal.
 4. A detector of dynamicgravitational force gradient fields, comprising: a body of elasticmaterial supported by a steel wire, said wire in turn being supported byan acoustically filtered support structure, said body comprising adistributed mass including a dynamic mass quadrupole moment responsiveto a dynamic gravitational force gradient having a characteristicfrequency, said response taking the form of relative accelerationbetween portions of said distributed mass, said body having a relativelyhigh Q longitudinal mechanical vibrational resonant mode excited by saidrelative acceleration, said mode having a nodal line at the position ofsaid wire; and electrical output means including a plurality of parallelconnected piezoelectric transducers attached to said body adjacent saidnodal line to form a relatively high Q electromechanical circuit that isresonant at said characteristic frequency for providing an outputsignal.
 5. A detector of dynamic gravitational force gradient fields,comprising: a body of elastic material supported by a steel wire, saidwire in turn being supported by an acoustically filtered supportstructure, said body comprising a distributed mass including a dynamicmass quadrupole moment responsive to a dynamic gravitational forcegradient having a characteristic frequency, said response taking theform of relative acceleration between portions of said distributed mass,said body having a relatively high Q mechanical vibrational resonantmode excited by said relative acceleration, said mode having a nodalline at the position of said said wire; and electrical output meansincluding a plurality of parallel connected piezoelectric transducersattached to said body adjacent said nodal line to form a relatively highQ electromechanical circuit which circuit includes a superconductingcoil encased in a superconducting shield disposed in a liquid heliumbath and a tuning capacitor tuned so that said circuit is resonant atsaid characteristic frequency for providing an output signal.